Hardy spaces of vector-valued functions: duality

Abstract

We prove here that the Hardy space of B B -valued functions H 1 ( B ) {H^1}(B) defined by using the conjugate function and the one defined in terms of B B -valued atoms do not coincide for a general Banach space. The condition for them to coincide is the UMD property on B B . We also characterize the dual space of both spaces, the first one by using B ∗ {B^{\ast }} -valued distributions and the second one in terms of a new space of vector-valued measures, denoted B M O ( B ∗ ) \mathcal {B}\mathcal {M}\mathcal {O}({B^{\ast }}) , which coincides with the classical BMO ⁡ ( B ∗ ) \operatorname {BMO} ({B^{\ast }}) of functions when B ∗ {B^{\ast }} has the RNP.